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Decision making

Which of the following 3 games of chance should you play?
1. Draw a card from a standard deck. If it is an ace, you win $5; if it is red, you win $1 ( a red ace wins $6); anything else, you lose $2.
2. Draw a card from a standard deck. If it is a club or a face card, you win $2.50; anything else, you lose $1.
3. Draw 2 cards, without replacement, from a standard deck. If they are of the same suit, you win $5; if it is a pair, you win $10; anything else, you lose $2.
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2 Answers

The way to figure this out is to calculate the expected value, and play the game with the highest expected value.  The expected value is the sum of all the probabilities of winning or losing times the respective amounts.  Losing bets are counted as negative amounts.
 
First game:
 
Probability of drawing a red ace from a standard deck is 2/52 = 1/26 = 0.0384165
Probability of drawing a red card that is not a red ace is 24/52 = 6/13 = 0.4615384
Probability of drawing a black ace from a standard deck is 2/52 = 1/26 = 0.0384165
Probability of drawing a black card that is not an ace is 24/52 = 6/13 = 0.4615384
 
Expected value = 0.0384165*$6.00 + 0.4615384*$1.00 + 0.0384165*$5.00 - 0.4615384*$2.00 =
 
0.230499 + 0.4615384 + .1920825 - 0.9230768 = -0.0389569
 
Second game:
 
Probability of drawing a club or a face card is 22/52 = 0.4230769
Probability of drawing anything else = 30/52 = 0.576923
 
Expected value = 0.4230769*$2.50 - 0.576923*$1.00 = 1.0576922 - 0.576923 = 0.4807692
 
Third game:
 
Probability of drawing two cards in a row from the same suit = 13/52 *  12/13 = 0.2307692
Probability of drawing a pair = 4/52 * 3/4 = 0.0576923
Probability of drawing anything else = 0.7115385
 
Expected value = 1.153846 + 0.576923 -1.423077 =  0.307692
 
SECOND GAME has the highest expected value and is positive, so that is the game to play.

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